Electrical impedance tomographyusing equipotential line method(A new approach to Inverse Conductivity Problem)(A new approach to Inverse Conductivity Problem)(A new approach to Inverse Conductivity Problem)(A new approach to Inverse Conductivity Problem)
June-Yub Lee (Ewha),
Ohin Kwon (Konkuk),
Jeong-Rock Yoon (RPI)
Physical Background(Forward) Conductivity Problem
Ω∂=∂∂=
ΩΩ∈
ongnuORfu
tsinxufindxxgivenFor..)(
,),(σ
0=∇⋅∇ uσ
Inverse Conductivity Problem
Ω∂=∂∂= ongnuANDfu
givenfromxFind )(σ
Ω∂),( gf
Impedance~Voltage/Current
Tomography
Electrical Impedance Electrical Impedance Electrical Impedance Electrical Impedance TomographyTomographyTomographyTomography (EIT)(EIT)(EIT)(EIT)
생체조직의저항률 (20-100KHz)
10
100
1000
10000
100000
체액
혈장
혈액
근육(횡
방향) 간팔근육
신경조직
심장근육
근육(종
방향) 폐
지방 뼈
최소값
최대값
[Ω-cm]
경혈 및 경락의 전기적 특성
가 설
경혈과 경락 전기저항이낮은 지점이다.
즉, 조직의저항률이 낮다.
A model problem with heart and lungA model problem with heart and lungA model problem with heart and lungA model problem with heart and lung
Methods of EIT/ICP
gnuuxuf kkMin =
∂∂=∇⋅∇−∑ ,0 where)( Objective 2
σσσ
σ
voltagefi =currentgi =
)(xσ
• Single Measurement⇒ minimum information
• Many Measurement⇒ full information(?) on σ
• Infinite Measurement⇒ mathematical theory
Numerical Obstacles for EIT/ICP
σσ
σ
σσ
σσσ
~better Find ,0 Solve Guss
,0 where)()( Objective)(2
→=∂∂=∇⋅∇→
=∂∂=∇⋅∇−∑ Ω
gnuuσ(x)
gnuuxuxf
LMin
• Strongly nonlinear:
• Ill-posed problem:
)()( 22 Ω∂∈→Ω∈ LuL σσ
σσσσ ~~ uu ≈→≈/
fugnuu −→=∂∂=∇⋅∇ σ
σσ σσ using ~better Find ,0 Solve
σσ on on restrictiu Regularity 0condition Stop →≅− fu
Ill-posed Nonlinear
• 50 iterations
EIT and MREIT(Magenetic Resonace Electrical Impedance Tomography)
0.95 mA
Current SourceVoltmeter
변수형 고속 영상복원 알고리즘
진단 변수 추출
심폐기능소화기능배뇨기능뇌기능
비정상 조직 검출기타 변수
실시간실시간영상영상 모니터링모니터링
MREIT고해상도 영상
적응형 요소융합및 세분화
변수형 고속알고리즘
초기해초기해초기해초기해
(1)문제정의
(2)자속밀도의 영상화
MRCDI 및 MREIT의 원리
1E2EΩ
∂Ω
( ), , ,Vρ J BI I
n3 \ Ω
L− L+
a
1 ( ) 0( )
Vρ
∇ ∇ =
r
r1 on V J
nρ∂ = ∂Ω∂
1( ) ( )( )
Vρ
= − ∇J r rr
03'( ) ( ') '
4 'J dvµ
π Ω
−= ×−∫
r rB r J rr r
30
3\
03
'( ) ( ') ( ') '4 '
'( ') ( ') '4 '
I
L
dv
I dl
µπ
µπ
±
±
Ω
−= ×−
−= ×−
∫
∫
r rB r J r a rr r
r rr a rr r
0
1( ) ( )B
µ= ∇×J r B r
1E
2E
x
yz
x yz
1E
2E
1E
2E
x
yz
Transversal ImagingSlice
Three Sagittal ImagingSlices
Three Coronal ImagingSlices
전류밀도 영상 실험용 팬텀
자속밀도 및 전류밀도 영상
y
z
]Tesla[
y
z
]Tesla[
x
z
]Tesla[
x
z
]Tesla[
x
y
]Tesla[
x
y
]Tesla[
Bx By
Bz
]mA/mm[ 2
x
y
]mA/mm[ 2 ]mA/mm[ 2
x
y
|J|0
1µ
= ∇×J B
자속밀도 및 전류밀도 영상
J-Substitution Algorithm
OriginalConductivity
ReconstructedConductivity
Current Density
Schematic Diagram for Equi-potential Line Method
Numerical Algorithm
Equipotential Line Method
Smooth conductivity distribution
Conductivity Result with noisy J
Phantom with 1%Add+10%Mul Noise
Artificially generated Human Body
1%Add+10%Mul Noise
Conclusions and further works• Fast, stable and efficient• Well-posed (error~noise)• Better interpolation method to handle discontinuous cases
• Noise handling for high conductivity ratio case where |J|~0
• 3D extension is trivial But 3D data for J is expensiveUsage of Bz instead of J is better
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